Relations in the cohomology ring of the moduli space of flat $SO(2n+1)$-connections on a Riemann surface

TL;DR

This paper studies the algebraic structure of the cohomology ring of the moduli space of flat $SO(2n+1)$-connections on a Riemann surface, constructing Chern classes and proving their relations vanish below the space's dimension.

Contribution

It introduces geometric representatives for Chern classes of line bundles over the moduli space and proves a vanishing relation generalizing Newstead's conjecture.

Findings

Constructed geometric representatives for Chern classes.

Proved the vanishing of the generated ring below the moduli space dimension.

Generalized a conjecture of Newstead regarding cohomology relations.

Abstract

We consider the moduli space of flat $SO(2n+1)$-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalising a conjecture of Newstead.